Investment without Coordination Failures

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# Investment without Coordination Failures 
### Brian C. Albrecht University of Minnesota Department of Economics
### January 31, 2020 Paper <a href="http://bit.ly/Albrecht-JMP">bit.ly/Albrecht-JMP</a> Slides <a href="http://bit.ly/Albrecht-JMP-Slides">bit.ly/Albrecht-JMP-Slides</a>

---

- Two pure-strategy, Nash equilibria

1. Pareto optimal, efficient investment: (Invest, Invest)
  2. Pareto dominated, **coordination failure**: (Don't Invest, Don't Invest)

- Both pure-strategy are stable; mixed-strategy is not

---
Competitive Coordination
====================================

- What if there are **many players** and they are **operating in a market**?

- These features matter because important real-world investments are sunk before entering a market with many players

- Invest in education before being hired
 - Produce goods before having a buyer

- When investments are sunk, *i.e.* when markets are incomplete, coordination failures can arise even with competitive markets
  - Makowski & Ostroy (1995), Cole, Malaith, & Postlewaite (1999a, 1999b), Makowski (2004), N&ouml;ldeke & Samuelson (2014), Felli & Roberts (2016)

--
  
- Previous literature only studied **existence of coordination failures**

---
My Question
====================================

- Are the market investments we observe likely to be inefficient?

- **Are coordination failures robust**?

- Should we predict coordination failures?

## My Answer

- **No**, if markets are competitive

- Formally, inefficient equilibria are not robust to errors in a trembling-hand refinement

`$\Rightarrow$` We should predict efficient market outcomes

---
Mechanism
====================================

- In competitive markets, players only care about **prices**

- To not invest, each entreprenuer must conjecture she would get paid less than 1 if she invests

- **Somebody must be wrong**

--
- Main Results:
  
  - Prop 1: Conjectures must contradict during any coordination failure

- Theorem 1: Contradictory conjectures are not robust to trembles
  - `$\Rightarrow$` Coordination failures are not robust

---
Related Literature
====================================

Non-cooperative before market/cooperative game:
Makowski & Ostroy (1995), Cole, Malaith, & Postlewaite (1999), Makowski (2004), Brandenburger & Stuart (2007), N&ouml;ldeke & Samuelson (2014)
 - Contribution: **focus on robust equilibria** via a refinement

---
Related Literature
====================================

Competitive equilibrium refinements: Gale (1992, 1996), Dubey & Geanakoplos (2002), Dubey, Geanakoplos, & Shubik (2004), Zame (2007), Scheuer & Smetters (2018)
 - Contribution: study **dynamic models with coordination failures**

---
Related Literature
====================================

Coordination failures are not robust: Albrecht (2016), Penta & Zuazo-Garin (2018)
 - Contribution: **Competition rules out** coordination failures

---
Road Map for Talk
====================================
1. **Example**
  
  - Environment
  - Equilibrium
  - Refinement
  
2. General Matching Model
  
  - Competitive Equilibria with Fixed Investments
  - Ex Post Efficiency

3. Investment Game where Players Choose Investments

- Investment Equilibria
  - Perfect Investment Equilibria
  - Ex Ante Efficiency

---
Sketch of General Environment
====================================

- Continuum of buyers and sellers, finite types

- Transferable utility

.center[![](https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_B.png)]

???
Simple set up to focus on the coordination aspect

Ignore hold-up, bargaining issues

---
Example
====================================

- Agents are endowed with a type `$t \in T = \{\beta, \sigma\}$`: "buyer" or "seller"

- Measure one of both

- Buyers choose an investment `$b \in B= \{0,1\}$`

- Sellers choose an investment `$s \in S = \{0,1\}$`

- Cost to buyer = `$\frac{1}{4}b$`

- Cost to seller = `$\frac{1}{4}s$`

- Surplus `$v(b,s) = bs$` of matching

---
Stage 2: Matching Market
===================================

- A buyer `$b$` chooses a **match contract** `$(b,s) \in \left\{(b,0) , (b,1)\right\}$` guaranteeing the right to join a match with `$s$`

- A seller `$s$` chooses `$(b,s) \in \left\{(0,s) , (1,s)\right\}$`

- Matches have competitive prices: `$p : B \times S \to \mathbb{R}$`

- `$p(b,s)$`: transfer from buyer with `$b$` to seller with `$s$`

- Matching market indirect utility for `$b$`:

`$$v^*_b(p) = \max_{s}\left\{ v(b,s) - p(b,s) \right\}$$`

- Matching market indirect utility for `$s$`:

`$$v^*_s(p) = \max_{b}\left\{ p(b,s) \right\}$$`

---
Stage 1: Decision Problem
==============================

- Agents only have **price conjectures**: `$\tilde{p}^t : B \times S \to \mathbb{R} \quad \forall t \in \{\beta,\sigma\}$`
  
  - Not a distribution but can think of conjectures = expected price
    
--

- Buyer's investment problem: `$$\max_{b} \left\{\underbrace{ v^*_b\left(\tilde{p}^\beta\right)}_{\text{Utility from Conjectured Optimal Contract}}  - \underbrace{\frac{1}{4}b}_{\text{Investment Cost}}\right\}$$`

- Seller's investment problem: `$\max_{s} \left\{ v^*_s\left(\tilde{p}^\sigma\right)  - \frac{1}{4}s\right\}$`

---
Investment Equilibrium
====================================

Equilibrium is a vector of matches, prices, and conjectures, such that

1. Each buyer chooses `$(b,s)$` to maximize utility, given `$\tilde{p}^\beta(b,s)$`

2. Each seller chooses `$(b,s)$` to maximize utility, given `$\tilde{p}^\sigma(b,s)$`

3. Prices clear matching markets
  - Mass of buyers choosing `$(b,s)$` = mass of sellers choosing `$(b,s)$`

4. Rational conjectures: conjectures are not contradicted by the data
  - If positive mass of agents choose contract `$(b,s)$`, conjectures agree with the posted price: `$\tilde{p}^\beta (b,s)=\tilde{p}^\sigma (b,s)= p(b,s)$`
  
  - If zero mass choose `$(b,s)$`, conjectures are not pinned down

???
Informal Definition

---
Payoffs
=====================================

.center[![](https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_G.png)]

--
- Prices and conjectures are equilibrium objects

- Rational conjectures: if a match `$(b,s)$` is chosen, `$\tilde{p}^\beta (b,s)= \tilde{p}^\sigma (b,s)= p(b,s)$`

- Allows disagreement for matches that do not occur

???
We need to find conjectures and prices

---

Constructing Equilibria
====================================

- Efficient equilibrium: everyone invests

- Suppose realized prices: `$p(1,1)= \frac{1}{2}$`
  - Conjectures: `$\tilde{p}^\beta = \tilde{p}^\sigma = 0$` otherwise

???
SLOW SLOW SLOW

---
Efficient Equilibrium Payoffs
=====================================

.center[![](https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_J.png)]

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Constructing Equilibria
====================================

- Efficient equilibrium: everyone invests

- Suppose realized prices: `$p(1,1)= \frac{1}{2}$`
  - Conjectures: `$\tilde{p}^\beta = \tilde{p}^\sigma = 0$` otherwise

- Coordination failure: no one invests

- Suppose realized prices: `$p(0,0)=0$`
  - Conjectures: 
      - Buyers: `$\tilde{p}^\beta (1,1) = 1$`
      - Sellers: `$\tilde{p}^\sigma(1,1) = 0$`

???
SLOW SLOW SLOW

---
Coordination Failure Payoffs
=====================================

.center[![](https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_K.png)]

- To prevent buyer deviating: `$\tilde{p}^\beta (1,1) \geq \frac{3}{4}$`

- To prevent seller deviating: `$\tilde{p}^\sigma(1,1) \leq \frac{1}{4}$`

- **Result**: Every coordination involves contradictory conjectures
  - `$\tilde{p}^\beta (1,1) \neq \tilde{p}^\sigma(1,1)$`

???
Only optimize, given prices.

Think deeper: Need to think price will be 1 and that will clear market

---
Refinement
=======================

- Multiplicity from a Nash-style equilibrium

- But the coordination failure requires contradictory conjectures

- To discipline conjectures and give predictive power, I introduce a mild refinement

- Trembling-hand perfection (Selten 1975)
  
  - Simple way to capture mistakes/experimentation

- Show which equilibria are robust/stable to small mistakes

???
Robert Lucas taught us to beware of theorists bearing free parameters

---
Perturbed Strategies
====================================

When defining trembles,

1. Symmetric trembling hand only at investment stage: 
  - Each investment must be chosen with positive probability `$\epsilon(b), \epsilon(s) >0$` by each buyer and seller

1. Law of large numbers: in the aggregate each investment must be chosen by at least  `$\epsilon(b)$` buyers and  `$\epsilon(s)$` sellers

- Definition: A perfect equilibrium is limit of some sequence of equilibria for `$\epsilon = (\epsilon(b),\epsilon(s)) \to (0,0)$`

- **Claim**: The unique perfect equilibrium *allocation* is efficient

???
Symmetric: all players and all actions have same tremble
---
Uniqueness Proof
==========

- With trembling hand, each `$b$` and `$s$` are played `$\Rightarrow$` `$\tilde{p}^\beta(b,s) = \tilde{p}^\sigma(b,s) = p(b,s)$`

- Prices are no longer a free parameter

- Step 1: Besides `$p(1,1)$` fix all other prices at 0
  - Verify ex post these prices are part of equilibrium

---
Deviating from `$(0,0)$`
================================================

- Step 2: For `$\epsilon >> 0$`

.center[![](https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_H.png)]

- For `$p(1,1) > \frac{1}{4}$`, all sellers choose `$(1,1)$`

- For `$p(1,1) < \frac{3}{4}$`. all buyers choose `$(1,1)$`

--
- `$\Rightarrow$` For any price, someone wants to deviate from `$(0,0)$`
  - `$(0,0)$` cannot be voluntarily chosen in an equilibrium

---
Supply and Demand for `$(1,1)$`
==========

Step 3: For any `$\epsilon >> 0$`, again some price `$p(1,1) \in \left[\frac{1}{4}, \frac{3}{4}\right]$` clears market

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.center[![](https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_I.png)]

- For every `$\epsilon \to (0,0)$`, the match `$(1,1)$` is the unique equilibrium strategy choice `$\blacksquare$`

- However, prices are not unique
- *e.g.* if equilibrium payoffs are `$\frac{1}{4}$`, other prices can be in `$\left[0, \frac{1}{4}\right)$`

- Indirectly proved no mixed strategy equilibrium (endogenous trembles)
  - This is different from 2-entrepreneur example

---
Results for the Example
==============================

1. Inefficiencies (coordination failures) can occur under competition

2. However, coordination failures require contradictory conjectures

3. Experimentation eliminates contradictions `$\Rightarrow$` no coordination failures

4. [All perfect, competitive equilibria are efficient](#fulltalk)

- These results hold in a general environment 
 
---
Extending the Results
================================

- **Main Theorem**: for a general class of matching models with investments, all *perfect* investment equilibria are efficient

- General finite types, `$\beta \in X,\sigma \in Y$`
  
  - General finite investment levels
  
  - General cost of investment that can depend on types, `$c\left(\beta, b\right)$` and `$c\left(\sigma, s\right)$`
  
  -  General surplus function `$v(b,s)$`

---
  Not *So* Simple
====================================

As in any paper, technical complications arise, *e.g.* I need to

1. Model an explicit competitive matching market, given fixed investments, compared to picking a match ex ante

1. Integrate the non-cooperative investment stage with the competitive matching market

1. Define a tremble and perfect equilibrium with a continuum of players

- [Example highlights the core ideas](#conclusions)

---
Structure of General Proof
================================

- General buyers' problem: `$\max_{b,s} v(b,s) - \tilde{p}^\beta (b,s) - c\left(\beta, b\right)$`

- General sellers' problem: `$\max_{b,s}  \tilde{p}^\sigma (b,s)  - c\left(\sigma, s\right)$`

- Total welfare, given measure over types `$\mathcal{E}(\beta), \mathcal{E}(\sigma)$`

`$$\sum_\beta \left[ \max_{b,s}  v(b,s) - \tilde{p}^\beta (b,s)- c\left(\beta, b\right)\right] \mathcal{E}(\beta)  +  \sum_\sigma \left[\max_{b,s}  \tilde{p}^\sigma (b,s)  - c\left(\sigma, s\right)\right]\mathcal{E}(\sigma)$$`
--

- Efficiency

`$$\max_{b,s}  \left\{  \sum_\beta \left[v(b,s)  - p(b,s)- c\left(\beta, b\right)\right]\mathcal{E}(\beta) +    
  \sum_\sigma\left[p(b,s) - c\left(\sigma, s\right)\right] \mathcal{E}(\sigma)\right\}$$`

- Since each agent is choosing both `$(b,s)$`, total welfare = efficiency  `$\blacksquare$`

---
name:fulltalk
Road Map for Talk
====================================
1. Example

2. **General Matching Model**

- Competitive Equilibria with Fixed Investments
  - Conditional Efficiency

3. Investment Game where Players Choose Investments

- Investment Equilibria
  - Perfect Investment Equilibria
  - Efficiency

---
Stage 1: Types
====================================

- General endowed types: `$t \in T$`

- Partitioned into

- Buyer endowed types: `$\beta \in X$`

- Seller endowed types: `$\sigma \in Y$`

- Economy: a positive measure on `$T$`: `$\mathcal{E} \in M_+(T)$`

---
Stage 1: Investments
====================================

- General investments: `$a \in A$`

- Partitioned into

- Buyer investments: `$b \in B$`
  
  - Seller investments: `$s \in S$`

- Type determines cost of acquiring investment `$$c : T \times A \to \mathbb{R} \cup {\infty}$$`

- `$c(t,a)$` is the cost to a type `$t$` of investment `$a$`

---
Stage 2: Matching Game Distribution
====================================

- Individual choices lead to distribution of investments `$\mu \in M_+(A)$`

- Not keeping track of `$M_+(T \times A)$`

- For any investment `$a$`, `$\mu(a)$` is the mass of individuals with investment `$a$`

- Buyers: `$\mu_b \in M_+(B)$`
  
  - Sellers: `$\mu_s \in M_+(S)$`

---
Stage 2: Matching Game
====================================

- To allow individuals to remain unmatched, let `$B^{\emptyset} \equiv B \cup \emptyset$` and `$S^{\emptyset} \equiv S \cup \emptyset$`

- Bounded value function: `$v : B^{\emptyset} \times S^{\emptyset} \to \mathbb{R}$`

- Competitive prices: `$p : B^{\emptyset} \times S^{\emptyset} \to \mathbb{R}$`
  - Normalize: `$p(b,\emptyset) \equiv p(\emptyset,s) \equiv 0$`

- Could generalize to `$v(\beta,b,\sigma,s)$`, if sufficient prices `$p(\beta,b,\sigma,s)$`

- To focus on investment choice and coordination failures, I don't allow this

---
Stage 2: Matching Game
====================================

- A matching is a measure `$$x \in M_+ (B^{\emptyset} \times S^{\emptyset})$$`

- A matching `$x$` is **feasible** for `$\mu$` if `$x(\emptyset,\emptyset)=0$`, and
`$$\sum_{s' \in S^{\emptyset}}x(b,s') = \mu(b)~~~\forall b$$`
`$$\sum_{b' \in B^{\emptyset}}x(b', s) = \mu(s)~~~\forall s$$`

---
Stage 2: Equilibrium
====================================

Definition: The pair `$(x,p)$` is an (ex post) **competitive equilibrium** for `$\mu$` if `$x$` is feasible for `$\mu$`,
--

- For each `$b \in  \text{ supp } \mu_b$` and each `$(b,s^*) \in \text{ supp } x,$`
the match maximizes `$b$`'s utility:

`$$s^* \in \text{ argmax }_{s \in \text{ supp }  \mu_s }\left\{ v(b,s) - p(b,s)\right\},$$`
--

- For each `$s \in  \text{ supp } \mu_s$` and each `$(b^*,s) \in \text{ supp } x$`, the match maximizes `$s$`'s utility:
--

`$$b^* \in \text{ argmax }_{s \in \text{ supp }  \mu_b } \left\{p(b,s)\right\}.$$`

???
Walk slowly through each

---
Stage 2: Efficiency
====================================

- Social matching gains function given `$\mu$`
`$$g(\mu) \equiv \text{max}_{x} \sum_{b \in B^{\emptyset}} \sum_{s \in S^{\emptyset}} v(b,s) x(b,s)~~~ s.t. x \text{ is feasible given } \mu$$`
- An allocation that attains `$g(\mu)$` is **conditionally efficient**

---
Stage 2: Efficiency
====================================
- **Conditional First Welfare Theorem**:  If a pair `$(x,p)$` is competitive for `$\mu$`, then it is conditionally efficient

- It is conditional because maximization only holds within the support

- `$\text{supp }\mu$` may not include the ex ante efficient `$b$` and `$s$`

- Given the choice in a non-cooperative setting, do people choose the efficient `$b$` and `$s$`?

---
Road Map for Talk
====================================
1. Example

2. General Matching Model
  
  - Competitive Equilibria with Fixed Investments
  - Conditional Efficiency

3. **Investment Game where Players Choose Investments**

- Investment Equilibria
  - Perfect Investment Equilibria
  - Efficiency

---
Stage 1: Investments
====================================

- Fix the ex ante population `$\mathcal{E}$`

- An allocation of investments is a measure `$\nu \in M_+(T\times A)$`
  - Marginals: `$\nu_{T}, \nu_A$`
  - `$\mu = \nu_{A}$`
  
- An allocation `$\nu$` is *feasible* for `$\mathcal{E}$` if `$\nu_T = \mathcal{E}$`

- Each agent of type `$t$` has price conjectures: `$\tilde{p}^t : B^{\emptyset} \times S^{\emptyset} \to \mathbb{R}$`

???
`$\nu$` is not v$
---
Conjectured Indirect Utility
====================================

- Define conjectured indirect utility of matching

- Buyer with `$b$` who conjectures `$\tilde{p}^\beta$`

`$$v^*_b\left(\tilde{p}^\beta\right) = \max_{s \in S^\emptyset}\left\{ v(b,s) - \tilde{p}^\beta(b,s)\right\}$$`
- Seller with `$s$` who conjectures `$\tilde{p}$`

`$$v^*_s\left(\tilde{p}^\sigma\right) = \max_{b \in B^\emptyset} \left\{\tilde{p}^\sigma(b,s)\right\}$$`

---
Investment Equilibrium
====================================

- A tuple `$\left(\nu,\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)$` is an **investment equilibrium** for `$\mathcal{E}$` if `$\nu$` is feasible, `$(x,p)$` is a competitive equilibrium for `$\nu_A$`,

Buyers optimize their investment: for all `$(t,b) \in \text{supp } \nu$`

`$$v_b^*(\tilde{p}^\beta) - c(t,b) \geq v^*_{b'}(\tilde{p}^\beta) - c(t,b') ~~~\forall b' \in B$$`

Sellers optimize: for all `$(s,t) \in \text{supp } \nu$`

`$$v_s^*(\tilde{p}^\sigma) - c(s,t) \geq v^*_{s'}(\tilde{p}^\sigma) - c(s',t) ~~~\forall s' \in S$$`

Rational conjectures: $$\tilde{p}^t(b,s) = p(b,s) ~~\forall (b,s) \in \text{ supp } \mu_b \times  \text{ supp } \mu_s $$

---
Efficiency
====================================

- Total cost of investments `$\nu$` is `$\sum_A \sum_T c(t,a) \nu(t,a).$`

- Total surplus from `$\nu$` is
`$$G(\nu) = g(\nu_{A}) - \sum_A \sum_T c(t,a) \nu(t,a).$$`

- The allocation `$\nu$` is unconditionally **efficient** for `$\mathcal{E}$` if it is feasible and `$G(\nu) \geq G(\nu')$` for all other feasible allocation `$\nu'$`

---
Unconstrained conjectures
====================================

- Without more structure, there will often be many equilibria

- Suppose not matching is generates no value generation: `$v(\emptyset,s)=v(b,\emptyset)=0$`
  - Often assumed in buyer/seller markets
  
- `$b=0, s=0$` can be part of an equilibrium through appropriate conjectures
  - No matter how large `$v(b,s)$` is, sellers can still rationally conjecture that `$p(b,s)=0$`

- Weakness of Nash equilibrium: 
  - In this class of matching games, the surplus maximizing **and minimizing** outcomes can occur

---
Need for Refinement
====================================

- Off-path conjectures are a free parameter

- Why would sellers conjecture `$\tilde{p}^t(1,1) = 0$`?

- Economists have recognized this issue in other competitive contexts:

- Zame (2007) - "imposing no discipline would admit equilibria which are **viable only because different agents hold contradictory conjectures**"

- Gale (1992) - "Typically, there exists a large number of equilibria and **some refinement of the equilibrium concept is required to give the theory predictive power**. One such refinement is based on the notion of the 'trembling' hand."

---
Perturbations
====================================

- Consider a perturbed strategy vector for all buyers, `$\epsilon_B = (\epsilon(b))_{b \in B},$` satisfying  `$\epsilon(b)>0$` for all `$b \in B$` and 
`$$\sum_B \epsilon(b) \leq 1$$`
- For all sellers `$\epsilon_S = (\epsilon(s))_{s \in S},$` satisfying  `$\epsilon(s)>0$` for all `$s \in S$` and 
`$$\sum_S \epsilon(s) \leq 1$$`

- A perturbed game is indexed by the vector of perturbed strategies `$\epsilon = (\epsilon_B,\epsilon_S)$`

---
`$\epsilon$`-Investment Equilibria
====================================

- An allocation `$\nu(\epsilon)$` is `$\epsilon$`-feasible for `$\mathcal{E}$` if  `$\nu_T = \mathcal{E}$` and for all `$a \in A$`
`$$\nu_{A}(\epsilon(a)) \geq \epsilon(a)$$`

- A tuple `$\left(\nu(\epsilon), \left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)$` is an `$\epsilon$`-investment equilibrium for `$\mathcal{E}$` if `$\nu$` is `$\epsilon$`-feasible, `$p$` is a competitive price for `$\nu_A$`, and for all `$(t,a)$` such that `$\nu_{A}(\epsilon) > \epsilon$`,

`$$v_a^*(\tilde{p}^t) - c(t,a) \geq v^*_{a'}(\tilde{p}^t) - c(a',t) ~~~\forall a' \in A$$`

- Note that by construction, `$\text{ supp } \nu_{A}(\epsilon) = A$`

- A tuple  `$\left(\nu,\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)$` is a **perfect investment equilibria** if there exists a sequence of `$\epsilon$`, such that `$\lim_{k \to \infty} M(\epsilon^k)=0$` such that `$\left(\nu(\epsilon),\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right) \to \left(\nu,\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)$`.

---
Restricted Efficiency
====================================
- The allocation `$\nu(\epsilon)$` is `$\epsilon$`-efficient for `$\mathcal{E}$` if it is feasible and `$G(\nu(\epsilon)) \geq G(\nu'(\epsilon))$` for all other `$\epsilon$`-feasible allocation `$\nu'$`

- **Lemma 1**: If `$(\nu(\epsilon), p)$` is an `$\epsilon$`-investment equilibrium, then it is `$\epsilon$`-efficient.

- Proof:

Let `$Q(\epsilon)$` be the utility generated by the trembling actions

`$$Q(\epsilon)=  \sum_b \left\{\sum_\beta  \left[v^*_b\left(\tilde{p}^\beta\right) - c(\beta,b)\right]\nu_T(\beta)\right\}\epsilon(b)$$`
 
`$$+ \sum_s \left\{\sum_\sigma  \left[v^*_s\left(\tilde{p}^\sigma\right) - c(\sigma,s)\right]\nu_T(\sigma)\right\}\epsilon(s).$$`

---
Proof (continued)
====================================

- Hence,

`$$\sum_\beta \left[ \max_{b,s}  v(b,s) - \tilde{p}^\beta (b,s)- c\left(\beta, b\right)\right] \nu_T(\beta)\left( 1 - \epsilon(b)\right)$$`

`$$+  \sum_\sigma \left[\max_{b,s}  \tilde{p}^\sigma (b,s)  - c\left(\sigma, s\right)\right]\nu_T(\sigma)\left( 1 - \epsilon(s)\right)$$`

$$ + \underbrace{Q(\epsilon)}_{\text{Constrained Choice}}$$

- But since all actions are played by trembles, `$\tilde{p}^\beta(b,s) =\tilde{p}^\sigma(b,s)$`

- Therefore they optimize the entire left expression

- It is exactly the complete market problem, which is efficient `$\qquad \blacksquare$`

---
Full Efficiency
====================================

- **Theorem**: If  `$\left(\nu,\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)$` is a perfect investment equilibrium, then it is efficient.

- Proof:

Immediate from Lemma 1, since  `$\underbrace{Q(\epsilon)}_{\text{Constrained Choice}} \to 0$` `$\qquad \qquad \qquad \qquad \blacksquare$`

---
Common Conjectures Justifications
====================================

- Predictive power comes from imposing more restrictions on conjectures than just rational conjectures

- Trembling hand + large market `$\Rightarrow$` as-if complete markets

- Complete markets means rational conjectures `$\Rightarrow$` common conjectures

- Other approaches:
  - Dubey and Geanakoplos (2002): fictitious seller who contributes an infinitesimal to each health insurance pool
  
  - Dubey, Geanakoplos, and Shubik (2005): government intervenes to sell infinitesimal quantities of each asset and fully delivers on its promises
  
  - Zame (2007): only considers "common-conjecture" equilibria

---
name:conclusions
Food for Applied Thought
====================================

- "Big push" policies attempts to move economies/sectors from coordination failure to more efficient outcome
  
  - Rosenstein-Rodan (1943), Murphy, Shleifer & Vishny (1988)
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- Idea: government policy can force investment, so other actors' best-reply is to also invest

- If works, very low cost way to increase output

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<img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz2_A.png">

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<img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz2_D.png">

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background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/jde.png)
background-size:contain
---
background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/jep.png)
background-size:contain

---
<img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz2_D.png">

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<img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz2_E.png">

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Conclusion
====================================

- Under competition, coordination failures rely contradictory conjectures

- If people can disagree, then of course there are many equilibria

- If we want predictive power, we must use a refinement

- I use trembling-hand perfection

Main theorem: Modified First Welfare Theorem

.center[Even with **incomplete and endogenous markets**,]
  .center[every perfect investment equilibrium is efficient.]

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class: center, middle

# Thank You

### Paper: [bit.ly/Albrecht-JMP](https://bit.ly/Albrecht-JMP)

### Slides: [bit.ly/Albrecht-JMP-Slides](https://bit.ly/Albrecht-JMP-Slides)